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While introducing myself to differential equations, I read that the solution to a differential equation may contain an arbitrary constant without being a general solution.

I have been solving initial value problems under the concept of anti-differentiation for a long time now. I am a little aware of general and particular solutions from my engineering classes. I have always thought that a solution having an arbitrary constant cannot be anything other than a general solution. I don't think a particular solution can have an arbitrary constant in it. The lines that I read today left me curious. I am really curious to see these equations and maybe know how they work.

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  • $\begingroup$ If you face a differential equation of order $n$, the general solution contains $n$ constants. Now, if you are given $m$ conditions $(m \leq n)$, $m$ of these constants "disappear" since fixed by the conditions. $\endgroup$ – Claude Leibovici Sep 10 '17 at 6:55
  • $\begingroup$ So, the solution, having some constants, is now a particular solution under a few conditions( m ), right? $\endgroup$ – R004 Sep 10 '17 at 6:58
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    $\begingroup$ I think what your text is trying to say is that a solution containing the right number of arbitrary constants, although it is a "general" solution (not a particular solution), yet it may fail to be the most general solution; i.e., there may be particular solutions which can not be obtained by assigning values to the arbitrary constants. $\endgroup$ – bof Sep 10 '17 at 7:01
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    $\begingroup$ For example, the (non-linear) first order equation $$\frac{dy}{dt}=y^2$$ has the "general solution" $$y=\frac1{C-t}$$ but it also has the "singular solution" $$y=0$$ which is not covered by the "general solution". $\endgroup$ – bof Sep 10 '17 at 7:05
  • $\begingroup$ This is a very interesting example. I appreciate it. It does justify your "most general solution" statement. But there still is a room for doubt about whether the text is exactly conveying what you just did. $\endgroup$ – R004 Sep 10 '17 at 7:13
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One can find examples (not exclusively) when the solution includes multi-valuated functions, which is the sometimes the case of inverse functions. For example, the ODE : $$\cos(y(x))\frac{dy}{dx}=1$$ has this family of solutions : $$y(x)=\sin^{-1}(x+c_1)$$ This solution includes an arbitrary constant $c_1$ but is not the general solution which is : $$y(x)=\sin^{-1}(x+c_1)+2\pi\:n$$

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  • $\begingroup$ This is a great example! $\endgroup$ – R004 Sep 10 '17 at 9:45
  • $\begingroup$ Lastly, the text mentioned that establishing that a solution "is" general requires deeper knowledge of the theory of differential equations. Although I am a beginner at differential equations, could I, if you could provide, take a peak at the procedure? $\endgroup$ – R004 Sep 10 '17 at 9:49
  • $\begingroup$ I am afraid that it is not possible to answer on a general manner. There is no general method to solve all kind of differential equations and so, for the generality of the solutions found in all cases. But they are some particular methods that you will progressively learn. $\endgroup$ – JJacquelin Sep 10 '17 at 11:00

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